Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.
problem Investigating the Funk-Finsler metric in spaces of constant curvature.
method Explicitly computed S-curvature, Riemann curvature, Ricci curvature, and flag curvature. result The S-curvature and flag curvature of the Funk-Finsler metric in hyperbolic, spherical, and Euclidean spaces are bounded. This paper aims to study the W-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time is semi-symmetric given that the W-curvature tensor is semi-symmetric whereas energy-momentum tensor T of a space-time having a divergence free W-curvature tensor is of Codazzi type. A space-t…
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
Study classifies zero mean curvature surfaces with planar curvature lines.
problem Characterizing surfaces with specific curvature properties.
method Complete classification and investigation of their relationship to Thomsen-type surfaces.
result Zero mean curvature surfaces with planar curvature lines belong to a 1-parameter family.
Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.
problem Geodesic curvature in Alexandrov spaces with curvature below.
method Comparison and rigidity theorems for geodesic curvatures.
result Generalized known results for geodesic curvature in spaces with curvature above.
The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and su…
Study on surfaces pinched by curvature in space forms converging under specific conditions.
problem Investigating convergence of surfaces pinched by curvature in space forms.
method Proving convergence theorems for surfaces pinched by normal curvature in 4-dimensional space forms.
result Generalizes Baker-Nguyen's convergence theorem for surfaces pinched by curvature.
The aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of Hp-scalar curvature and of Hp-constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of Hp-scalar curvature to be of perpendicular scalar curvature i…
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
Paper proves curvature conditions are preserved in metric spaces.
problem Preserving curvature conditions in metric spaces.
method Doubling and gluing constructions to preserve RCD(K,N) condition. result Proves RCD(K,N) condition is preserved. The paper examines biconservative hypersurfaces with constant curvature in space forms.
problem Characterizing biconservative hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces with four distinct principal curvatures in space forms.
result Every biconservative hypersurface has constant mean and scalar curvature.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.
problem Calculating curvatures in homogeneous Finsler spaces.
method Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
result Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.
Study 2D spaces with curvature, focusing on structure and approximations.
problem Characterize and understand 2D metric spaces with curvature constraints.
method Lipschitz homotopy approximations, curvature measures, convergence analysis.
result Established Gauss-Bonnet Theorem and characterized spaces.
Proves curvature tensor convergence for smoothable spaces.
problem Curvature tensor behavior in smoothable Alexandrov spaces.
method Weak convergence of curvature tensors in noncollapsing sequences.
result Proves convergence of curvature tensors in smoothable Alexandrov spaces.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
Paper estimates curvature of semi-convex solutions in hyperbolic space.
problem Curvature estimation for semi-convex solutions in hyperbolic space.
method Used concavity inequality for Hessian operator.
result Established curvature estimates for semi-convex solutions and admissible solutions.
The study finds non-trivial elements in moduli spaces of curvature metrics.
problem Identifying non-trivial elements in moduli spaces of curvature metrics.
method Constructing elements in homotopy groups of moduli spaces of positive curvature metrics.
result Non-trivial elements found in moduli spaces of positive curvature metrics.
New findings on hypersurfaces with specific curvature properties in space forms.
problem Characterizing hypersurfaces with almost constant curvature in space forms.
method Analyzing starshaped hypersurfaces with various curvature conditions.
result Closed starshaped hypersurfaces with almost constant mean curvature or higher order mean curvature are close to geodesic spheres.
Curvature of Bazaikin spaces fully characterized, with unique quasi-positive metric.
problem Characterizing sectional curvature of Bazaikin spaces.
method Completely characterized the sectional curvature of all 13-dimensional Bazaikin spaces.
result Unique quasi-positively curved Riemannian metric on Bazaikin spaces, with a unique almost positively curved but not positively curved space.
Classifies hypersurfaces with constant curvature in product spaces.
problem Classifying hypersurfaces with constant curvature in product spaces.
method Analyzing hypersurfaces in R^k x S^{n-k+1} and R^k x H^{n-k+1} for 2 <= k <= n-1.
result Complete description of hypersurfaces with constant curvature in product spaces of space forms.
New theorem links quaternionic-Kähler manifolds to symmetric spaces.
problem Understanding curvature properties of quaternionic-Kähler manifolds.
method Analyzing positive scalar curvature and nonnegative sectional curvature.
result Compact quaternionic-Kähler manifolds with these properties are symmetric.
Paper proves curvature estimate for curved spaces.
problem Estimating total curvature of closed hypersurfaces.
method Used non-positively curved symmetric spaces.
result Proved total curvature estimate and isoperimetric inequality.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Curvature of 2D subsets preserved in their space.
problem Understanding curvature of subsets in 2D spaces.
method Analyzing subsets with vanishing first homology.
result Closed subsets inherit curvature bounds from ambient spaces.
Study on spaces of metrics with intermediate curvature bounds.
problem Understanding spaces of metrics with lower bounds on intermediate curvatures.
method Analyzing spaces of Riemannian metrics with specific curvature bounds on high-dimensional Spin-manifolds.
result Spaces of metrics with positive p-curvature and k-positive Ricci curvature have non-trivial homotopy groups.
The paper derives curvature inequalities for submersions from quaternionic space forms.
problem Deriving curvature inequalities for submersions from quaternionic space forms.
method Analyzing Ricci and scalar curvatures of horizontal and vertical distributions in anti-invariant submersions.
result Established Ricci curvature inequality for anti-invariant submersions.
The paper classifies surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.
problem Classifying surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.
method Analyzing conditions equivalent to constant principal curvature, mean curvature, and second mean curvature.
result Surfaces of L1-2-type in De Sitter and anti De Sitter spaces are either standard products, scrolls, or have non-constant curvature properties. Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least 1/2n(n−1)+1, for almost all val…
In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly …
The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.
Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.
problem Preserving positive sectional curvature in contracting curvature flows in hyperbolic space.
method Homogeneous speed flow with positive sectional curvature, including kth mean curvature flow. result Positive sectional curvature is preserved and the hypersurface contracts to a round point in finite time.
The paper examines curvature properties of twistor spaces.
problem None explicitly stated in the abstract.
method Review of Riemannian and almost Hermitian geometry of twistor spaces.
result Curvature properties of twistor spaces are explored.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Curvature study on Eschenburg spaces with two metrics.
problem Characterizing and studying curvature of Eschenburg spaces.
method Investigation of curvature with respect to two metrics: Eschenburg and Wilking.
result For Wilking's metric, most examples have open sets of zero-curvature planes.
Abstract: Characterizes spaces with positive scalar curvature.
problem Spaces with positive scalar curvature and invariant metrics.
method Cohomogeneity one manifolds and homogeneous spaces with compact Lie group actions.
result Characterization of spaces with positive scalar curvature.
We prove that Berwald spaces whose flag curvature is nowhere vanishing are in fact Riemannian spaces. This means that any Berwald space with flag curvature bounded below by a positive number must be also Riemannian. This rigidity result shows the importance of non-Riemannian examples when imposing flag curvature bounds…
The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
Study submanifolds in curved spaces with specific curvature bounds.
problem Investigate submanifolds in curved spaces with Ricci pinched conditions.
method Analyze submanifolds in Riemannian space forms with a lower Ricci curvature bound.
result Eliminate the need for mean curvature vector field to be parallel.
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
The study examines symmetries in spaces with positive or non-negative curvature.
problem Understanding symmetries in spaces with curvature constraints.
method Survey of existing results for Riemannian manifolds with specified curvature properties and symmetries.
result Results on symmetries in spaces with curvature bounds.
The paper studies how convex hypersurfaces evolve under curvature flows in space forms.
problem Understanding the evolution of convex hypersurfaces under curvature flows in different space forms.
method Flow by powers of the Gauss curvature in space forms.
result Convex hypersurfaces under the flow by powers of the Gauss curvature in space forms contract to a point in finite time or converge to geodesic spheres.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
The study of curvature in spacelike submanifolds in pseudo-hyperbolic spaces.
problem Curvature properties of spacelike submanifolds in pseudo-hyperbolic spaces.
method Analyzing scalar and Ricci curvatures, proving bounds and characterizing submanifolds.
result Nonpositive scalar curvature for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.
Study on extremal subsets in geodesically complete spaces with curvature constraints.
problem Characterizing extremal subsets in GCBA spaces.
method Introduced and analyzed extremal subsets in GCBA spaces, proving their properties.
result Set of topological singularities forms an extremal subset under additional assumptions.